Adding a certain amount (x) of a reagent to a chemical solution, then measuring the concentrations of several different reaction products (should be g(x), actual measured values have some error). I want to know if the correct amount (x0) of the reagent was added.

I have a function f: ℝ→ℝ n that takes one input and produces n outputs. It represents the result of a physical process, and is equal to a theoretical model g(x) plus a measurement error. For a given sample, I can measure the values of f(x), and I want to find out the value of x. In particular, there...

For mathematics, you will definitely want a solid grasp of linear algebra and differential equations. A significant part of physics comes down to partial differential equations, which can get…difficult. On the more theoretical side, you may want to learn about topology and manifolds. And, in light o...

It’s worth noting that ambiguity is *also* possible when an Oxford comma is included: “He thanked his mother, Athena, and his training staff.” If the second comma were removed, the sentence would have only one meaning. …and then there are sentences which are ambiguous irregardful of whether a second...

Thanks for the suggestion! I'm not sure how to get around the fact that x and y are in vector spaces of different dimension, though. I know that invertible transformations do exist, but I couldn't even begin to figure out how to construct T(t) which is invertible for t>0, but approaches something l...

Form a continuous family of transformations T(t) such that T(0) = C, and T is invertible when t≠0. For example, I think T(t) = t·I + (1−t)·C should work. Find the general solution in terms of T when t≠0, then take the limit as t→0.

If you beat Advent.exe you get an option to save the image. I'm not sure of any way to link directly to the result, so I'm going to imgur this. The URL from which the image downloaded for me is https://xkcd.com/1975/v6xso1_right_click_save.png . Not sure if that is a temporary address or permanent ...

83.1 ~ 83.33... = 5/6 Take it up to 4 and see what happens? Guessing what's going on from one data point is going to be pretty impossible. I’ll try that when I have time to update the code. I’m trying to find an analytical solution. With n IID normal variates, their position in ℝ n is spherically-s...

Here’s the scenario: • You have a book, and some of the words are underlined. • I hand you another book. • You have to figure out if it is possible to start with your book, erase some of the non-underlined words, and end up with my book (or at least, the same words in the same order as my book). The...

https://imgs.xkcd.com/comics/memorable_quotes.png Mouseover: “"Since there's no ending quote mark, everything after this is part of my quote. —Randall Munroe” • • • Most xkcd comics have all text majuscule, but there is a lowercase letter in this one! Also, I’m probably going to use a few of t...

Okay, try this on for size: Can you prove that, for every positive integer n there exists a positive real number a n such that, if x is within distance a n of 0 then your function’s magnitude is less than the magnitude of x n ? In other words, by choosing x “close enough” to zero, can you show that ...

↶ One degree is approximately 1.75 percent In what way? It's not 1.75 percent of a circle, or even of a quarter-arc. (Vaguely related, 1px in CSS is about 1.25 arcminutes - the CSS length units are technically angle units, since they scale by viewing distance to subtend the same fraction of your vi...

I’ve been tinkering with the α·max + β·min approximation to the distance formula. As a quick refresher, if you know the x and y distances to some point and want to estimate the straight-line distance, you can take the dot product of ⟨x, y⟩ with an appropriately chosen vector ⟨α, β⟩. Since the dot pr...

Given a basis for a hyperplane in ℝⁿ, what’s the best way to obtain a vector orthogonal to it? In particular, given n independent vectors in ℝⁿ, is there an efficient way to calculate a vector orthogonal to the hyperplane containing all of their differences ( v i − v 0 )? The general problem would b...

As another minor thing, you can improve speed a little by looking at distance^2 rather than distance. The square root operation is one of the slowest basic operations a computer can do. Things often go quite a bit faster if you check whether distance^2<R^2 rather than doing the square root to calcu...

I mean, submitting a middling score on someone necessarily means having less impact on that person's chances than giving a 0 or a 5, right? Voters tend to have their preferences decided and want to vote strategically to the advantage of the candidates they prefer and the disadvantage of alternative...

Cueball knows what’s up, approval voting is *way* better than IRV, and Condorcet methods are generally better as well. There’s a new method being proposed in Oregon called “star voting” (aka. score-runoff) where you rate each candidate on a 0–5 scale (like website reviews) and the 2 highest-scoring ...

In either order there is a double plosive from the final consonant of the first word and the initial consonant of the second. This makes is a fairly awkward phrase to say out loud, especially when speaking quickly, so perhaps people avoid it for that reason. I’m not convinced this explains it. Afte...

I was reading about how adjectives in English have a certain order in which they appear (shown in this list ), and it occurred to me that “good big” is a pairing which obeys the ordering rule, yet nonetheless does not naturally appear in English (at least apart from compound-word situations like “bi...

Depending on exactly what you are doing, and how the data are organized, and how many data points there are, the following approaches may be beneficial: • Split the universe into spatial bins (small boxes, like a 3D checkerboard) by coordinates, and keep track of which atoms are in which bins. This ...

Are you also opposed to sort() returning a sorted array? Will you argue that arrays are an unsorted type, and it's valid for sort() to return an unsorted array, and it's the caller's job to handle that? Well if you’re working with an array of IEEE–754 floating-point values, some of which are NaN, a...